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PTEP 2015, 033B04 M.A. Paracha et al.
The hadronic functions τ
1
μ
and τ
2
μ
are given by
τ
1
μ
= A
q
2
γ
μ
+ iB
q
2
σ
μν
q
ν
+ C
q
2
q
μ
− D
q
2
γ
μ
γ
5
− iE
q
2
σ
μν
q
ν
γ
5
− F
q
2
q
μ
γ
5
(14)
τ
2
μ
= G
q
2
γ
μ
+ iH
q
2
σ
μν
q
ν
+ I
q
2
q
μ
− J
q
2
γ
μ
γ
5
− iK
q
2
σ
μν
q
ν
γ
5
− L
q
2
q
μ
γ
5
.
(15)
The auxiliary functions from A
q
2
to L
q
2
given in Eqs. (14)and(15) contains both short- and
long-distance effects, which are encapsulated in terms of Wilson coefficients and form factors. The
explicit for m of these functions can be written as follows:
A
q
2
= C
tot
9
f
1
q
2
−
2m
b
q
2
C
eff
7
f
T
1
q
2
B
q
2
= C
tot
9
f
2
q
2
−
2m
b
q
2
C
eff
7
f
T
2
q
2
C
q
2
= C
tot
9
f
3
q
2
−
2m
b
q
2
C
eff
7
f
T
3
q
2
D
q
2
= C
tot
9
g
1
q
2
−
2m
b
q
2
C
eff
7
g
T
1
q
2
E
q
2
= C
tot
9
g
2
q
2
−
2m
b
q
2
C
eff
7
g
T
2
q
2
(16)
F
q
2
= C
tot
9
g
3
q
2
−
2m
b
q
2
C
eff
7
g
T
3
q
2
G
q
2
= C
tot
10
f
1
q
2
H
q
2
= C
tot
10
f
2
q
2
I
q
2
= C
tot
10
f
3
q
2
J
q
2
= C
tot
10
g
1
q
2
K
q
2
= C
tot
10
g
2
q
2
L
q
2
= C
tot
10
g
3
q
2
.
The matrix element for the decay
b
→
+
−
given in Eq. (13) is useful to calculate the physical
observables. The formula for the double differential decay rate can be written as
d
2
b
→
+
−
d cos θ ds
=
1
2M
3
2β
√
λ
8π
3
M
tot
2
, (17)
where β ≡
1 −
4m
2
s
and λ =λ
M
b
, M
, s
≡ M
4
b
+ M
4
+s
2
−2M
2
b
M
2
−2sM
2
b
−2sM
2
.
Also, s is the square of the momentum transfer q and θ is the angle between the lepton and final-state
baryon in the rest frame of
b
. By using the expression of amplitude given in Eq. (13) and integration
over cos θ , one can get the expression of the dilepton invariant mass spectrum as
d
b
→
+
−
ds
=
G
2
F
α
2
EM
β
√
λ
2
13
π
5
M
3
b
V
tb
V
∗
ts
2
M
1
, (18)
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